- Appleton-Hartree equation
The Appleton-Hartree equation, sometimes also referred to as the Appleton-Lassen equation is a mathematical expression that describes the
refractive indexfor electromagnetic wavepropagation in a cold magnetized plasma. The Appleton-Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartreeand K. Lassen.
The equation is typically given as follows [ Citation | last = Helliwell | first = Robert | date = 2006 | title = Whistlers and Related Ionospheric Phenomena | edition = 2nd | publication-place = Mineola, NY | publisher = Dover | pages = 23-24 ] :
Definition of Terms
= complex refractive index
= electron collision frequency
= wave frequency
permittivity of free space
permeability of free space
= angle between the ambient
magnetic fieldvector and the wave vector
Modes of Propagation
The presence of the sign in the Appleton-Hartree equation gives two separate solutions for the refractive index [ Citation | last = Bittencourt | first = J.A. | date = 2004 | title = Fundamentals of Plasma Physics | edition = 3rd | publication-place = New York, NY | publisher = Springer-Verlag | pages = 419-429 ] . For propagation perpendicular to the magnetic field, i.e., , the '+' sign represents the "ordinary mode," and the '-' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., , the '+' sign represents a left-hand circularly polarized mode, and the '-' sign represents a right-hand circularly polarized mode. See the article on electromagnetic electron waves for more detail.
Propagation in a Collisionless Plasma
If the wave frequency of interest is much smaller than the electron collision frequency , the plasma can be said to be "collisionless." That is, given the condition
so we can neglect the terms in the equation. The Appleton-Hartree equation for a cold, collisionless plasma is therefore,
Quasi-Longitudinal Propagation in a Collisionless Plasma
If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., , we can neglect the term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton-Hartree equation becomes,
References;Citations and notes
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